Theorem onunisuc 2354

Description: An ordinal number is equal to the union of its successor.

Hypothesis

Ref	Expression
on.1	⊢ A ∈ On

Assertion

Ref	Expression
onunisuc	⊢ ∪suc A = A

Proof of Theorem onunisuc

Step	Hyp	Ref	Expression
1		on.1	. . 3 ⊢ A ∈ On
2	1	ontrc 2344	. 2 ⊢ Tr A
3	1	elisseti 1355	. . 3 ⊢ A ∈ V
4	3	unisuc 2299	. 2 ⊢ (Tr A ↔ ∪suc A = A)
5	2, 4	mpbi 164	1 ⊢ ∪suc A = A

Syntax hints:   = wceq 1091   ∈ wcel 1092  ∪cuni 1919  Tr wtr 2041  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  limsuclem 2360  tz7.44-2 2967

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

metamath.org